A well-posedness theory for second-order hyperbolic systems in bounded domains that Laplace-transforms in time and Fourier-transforms in the tangential directions, then rewrites the n-equation second-order system as a first-order system of exactly 2n pseudo-differential equations - avoiding the extra equations and side conditions that larger first-order reductions require. Students learn to locate the generalized eigenvalues of the transformed boundary-value problem and read off from them which boundary phenomena a given boundary condition admits: surface waves that decay away from the boundary, glancing waves that are constant normal to it, and oscillatory modes that lose a derivative at every reflection. The resulting classification into Strongly Boundary Stable, Boundary Stable, Stable and Unstable problems tells you whether a boundary condition yields a well-posed problem, how much regularity the solution gains or loses, and whether the problem can be localized and extended to variable coefficients.
This paper develops a comprehensive well-posedness theory for initial-boundary value problems of second-order systems in bounded domains, addressing boundary phenomena such as glancing and surface wa…